Then \(H^{\gamma }_p\)  is a Banach space with the given norm and
\(C^{\infty }_0(\mathbb {R}^d;\mathbb {R}^{d_1})\)  is dense in \(H^{\gamma }_p\)  (see [1]}). Note that \(H^{\gamma }_p\)  are
usual Sobolev spaces for \(\gamma =0,1,2,\ldots \) . It is well known that
the first order differentiation operator, \(D:H^{\gamma }_{p}\rightarrow H^{\gamma -1}_p\) , is bounded. On the other hand, if \(\text{supp}\,(u) \subset (a,b)\) , where \(-\infty <a<b<\infty \) , then
\(\Vert u\Vert _{H^{\gamma }_p}\le c(d,a,b)\Vert u_x\Vert _{H^{\gamma -1}_p}\) 
